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Problem 972

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Hyperbolic Plane

The hyperbolic plane can be represented by the open unit disc, namely the set of points $(x, y)$ in $\Bbb R^2$ with $x^2 + y^2 \lt 1$.

A geodesic is defined as either a diameter of the open unit disc or a circular arc contained within the disc that is orthogonal to the boundary of the disc.
The following diagram shows the hyperbolic plane with two geodesics; one is a diameter and the other is a circular arc.

Let $\mathcal V(N)$ be the set of points $(x, y)$ such that $x^2 + y^2 \lt 1$ and $x, y$ are both rational numbers with denominator not exceeding $N$.

Let $T(N)$ be the number of ordered triples $(P, Q, R)$ such that $P, Q, R$ are three different points in $\mathcal V(N)$ and there is a hyperbolic line passing through all of them.

For example, $T(2) = 24$ and $T(3) = 1296$.

Find $T(12)$.

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双曲平面

双曲平面可以用开单位圆盘来表示，也即$\Bbb R^2$中满足$x^2 + y^2 \lt 1$的点$(x, y)$所构成的集合。

定义测地线为开单位圆盘的直径或圆盘内与圆盘边界正交的圆弧。
如下图展示了双曲平面上的两条测地线；其中一条是直径，另一条是圆弧。

记$\mathcal V(N)$为满足$x^2 + y^2 \lt 1$且$x, y$均为分母不超过$N$的有理数的点$(x, y)$所构成的集合。

记$T(N)$为满足以下条件的有序三元组$(P, Q, R)$的数目：其中$P, Q, R$是$\mathcal V(N)$中三个不同的点，且存在一条双曲线（译注：即测地线）同时经过这三个点。
例如，$T(2) = 24$，$T(3) = 1296$。

求$T(12)$。

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